On fundamental groups related to the Hirzebruch surface F_1

TL;DR

This paper investigates the fundamental groups associated with the Hirzebruch surface F_1, demonstrating they are nearly solvable and supporting conjectures about degeneratable surfaces through braid monodromy invariants.

Contribution

It shows that the fundamental groups of the Hirzebruch surface F_1 are almost-solvable, extending understanding of their topological properties and stability under deformation.

Findings

Fundamental groups are almost-solvable.

Supports conjecture on degeneratable surfaces.

Provides explicit computation of braid monodromy.

Abstract

Given a projective surface and a generic projection to the plane, the braid monodromy factorization (and thus, the braid monodromy type) of the complement of its branch curve is one of the most important topological invariants, stable on deformations. From this factorization, one can compute the fundamental group of the complement of the branch curve, either in C^2 or in CP^2. In this article, we show that these groups, for the Hirzebruch surface F_{1,(a,b)}, are almost-solvable. That is - they are an extension of a solvable group, which strengthen the conjecture on degeneratable surfaces.